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The better pi

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tau

The better π

Beauty is the first test: there is no permanent place in this world for ugly mathematics.

—G.H. Hardy

In 2001, Bob Palais wrote the article π is wrong!, remarking on the fact that when π occurs in nature, it occurs as 2π most of the time. He suggests that 2π = 6.283185... should be given a name (τ is now a fan favorite) and that it should be used instead of π. Some found his arguments convincing, including Michael Hartl. In 2010, Hartl published The Tau Manifesto, and the τ movement was born.

Scientific articles and blog posts

Articles against τ:

News articles and blog posts

Click to expand

Other articles

Videos

Textbooks

In pop culture

Historical uses

Al-Kashi (1424)

(All quotes are translated from the German translation by P. Luckey, 1950.)

While Archimedes was able to bound the ratio of a circle circumference and diameter between 223/71 (≈ 3.1408) and 22/7 (≈ 3.1428), Jamshid al-Kashi wanted to determine it to a much higher precision. In his 1424 Treatise on the circumference of the circle, it was his goal that

in a circle whose diameter is 600,000 times the diameter of the earth, the difference between it [the calculated circumference] and the true circumference is less than a single hair, which is one sixth of the width of an average barley grain, such that it [the difference] in a [circle] which is smaller than that doesn't matter.

This translates to approximately 14 decimal digits.

His calculations are performed in sexagesimal (base 60) digits. In section 8, Transformation of the value of the circumference into the Indian digits under the condition that half of the diameter be one, he gives the result in decimal digits:

We put the digits from left to right onto a half-verse, to get a verse:

wa baḥǧā ḥahǧi ṣaz a za ṭah ḥawahu

muḥīṭun li-quṭrin huwa ’ṯnāni minhu

6 2 8 3 1 8 5 3 0 7 1 7 9 5 8 6 5

is the circumference for a diameter which is two thereof.

Further reading:

π in the times of Euler

The section Adoption of the symbol π of the excellent Wikipedia article Pi says:

In the earliest usages, the Greek letter π was used to denote the semiperimeter (semiperipheria in Latin) of a circle and was combined in ratios with δ (for diameter or semidiameter) or ρ (for radius) to form circle constants. (Before then, mathematicians sometimes used letters such as c or p instead.) The first recorded use is Oughtred's "δ.π", to express the ratio of periphery and diameter in the 1647 and later editions of Clavis Mathematicae. Barrow likewise used "$\frac{\pi}{\delta}$" to represent the constant 3.14..., while Gregory instead used "$\frac{\pi}{\rho}$" to represent 6.28...

The earliest known use of the Greek letter π alone to represent the ratio of a circle's circumference to its diameter was by Welsh mathematician William Jones in his 1706 work Synopsis Palmariorum Matheseos; or, a New Introduction to the Mathematics. The Greek letter first appears there in the phrase "1/2 Periphery (π)" in the discussion of a circle with radius one. However, he writes that his equations for π are from the "ready pen of the truly ingenious Mr. John Machin", leading to speculation that Machin may have employed the Greek letter before Jones. Jones' notation was not immediately adopted by other mathematicians, with the fraction notation still being used as late as 1767.

Euler started using the single-letter form beginning with his 1727 Essay Explaining the Properties of Air, though he used π = 6.28..., the ratio of periphery to radius, in this and some later writing. Euler first used π = 3.14... in his 1736 work Mechanica, and continued in his widely-read 1748 work Introductio in analysin infinitorum (he wrote: "for the sake of brevity we will write this number as π; thus π is equal to half the circumference of a circle of radius 1"). Because Euler corresponded heavily with other mathematicians in Europe, the use of the Greek letter spread rapidly, and the practice was universally adopted thereafter in the Western world, though the definition still varied between 3.14... and 6.28... as late as 1761.

In programming

Inclusion of a constant tau was rejected by the following projects:

Some equations

Trigonometry

$$\begin{split} \sin(\alpha) &= \sin(\alpha + \textcolor{orange}{\tau}) \quad\forall \alpha\\\ \cos(\alpha) &= \cos(\alpha + \textcolor{orange}{\tau}) \quad\forall \alpha\\\ \tan(\alpha) &= \tan(\alpha + \textcolor{orange}{\tau}) \quad\forall \alpha \end{split}$$

With τ being a full revolution, the following identities are very easy to grasp (for integers n). Remember the sine is the projection of the angle onto the y-axis, the cosine is the projection onto the x-axis.

$$\begin{alignat*}{3} \sin(n \textcolor{orange}{\tau}) &= 0, &\qquad \cos(n \textcolor{orange}{\tau}) &= 1,\\\ \sin((n + 1/4) \textcolor{orange}{\tau}) &= 1, &\qquad \cos((n + 1/4) \textcolor{orange}{\tau}) &= 0,\\\ \sin((n + 1/2) \textcolor{orange}{\tau}) &= 0, &\qquad \cos((n + 1/2) \textcolor{orange}{\tau}) &= -1,\\\ \sin((n + 3/4) \textcolor{orange}{\tau}) &= -1, &\qquad \cos((n + 3/4) \textcolor{orange}{\tau}) &= 0 \end{alignat*}$$

Trigonometric values off the grid can easily be estimated:

  • sin(27.4 π) – Where is my calculator?
  • sin(13.7 τ) – 13 full revolutions: forget about those. Plus .7, that's almost 3/4 of a revolution, so probably something close to −1. (Actual value: −0.95105651629...)

Surface area of the n-dimensional unit sphere

$$|U_n| = \frac{2\textcolor{teal}{\pi}^{n/2}}{\Gamma(n/2)} = \begin{cases} 2 & \text{if } n = 1\\\ \textcolor{orange}{\tau} & \text{if } n = 2\\\ |U_{n-2}| \times \textcolor{orange}{\tau} / (n - 2) & \text{otherwise} \end{cases}$$

n-dimensional Gegenbauer integral over the unit ball Sn

$$|G_n^{\lambda}| = \int_{S^n} \left(1 - \sum_{i=1}^n x_i^2\right)^\lambda = \begin{cases} 1&\text{for $n=0$}\\\ B\left(\lambda + 1, \frac{1}{2}\right)&\text{for $n=1$}\\\ |G_{n-2}^{\lambda}|\times \textcolor{orange}{\tau} / (2\lambda + n) & \text{otherwise} \end{cases}$$

Note that the Beta function B, with one argument ½, includes a factor $\sqrt{\pi}$.

Special cases:

  • Volume of the n-dimensional unit ball (λ = 0):

    $$|S_n| = \begin{cases} 1 & \text{if } n = 0\\\ 2 & \text{if } n = 1\\\ |S_{n-2}| \times \textcolor{orange}{\tau} / n & \text{otherwise} \end{cases}$$

  • The area of a disk (λ = 0, n = 2)

    $$|S_n(r)| = \frac{\textcolor{orange}{\tau}}{2} r^2 = \textcolor{teal}{\pi} r^2$$

  • n = 1, λ = −1/2

    $$\int_{-1}^1 \frac{1}{\sqrt{1-x^2}} = \textcolor{teal}{\pi}$$

  • n = 1, λ = 1/2

    $$\int_{-1}^1 \sqrt{1-x^2} = \frac{\textcolor{teal}{\pi}}{2}$$

n-dimensional generalized Cauchy volume (2λ > n)

As appearing in its one-dimensional version in the Cauchy distribution and Student's t distribution.

$$\begin{align*} |Y_n^{\lambda}| &= \int_{\mathbb{R}^n} \left(1 + \sum_{i=1}^n x_i^2\right)^{-\lambda}\\\ &= \begin{cases} 1&\text{for $n=0$}\\\ B\left(\lambda - \frac{1}{2}, \frac{1}{2}\right)&\text{for $n=1$}\\\ |Y_{n-2}^{\lambda}|\times \textcolor{orange}{\tau} / (2\lambda - n) & \text{otherwise} \end{cases} \end{align*}$$

Note again that the Beta function B, with one argument ½, includes a factor $\sqrt{\pi}$. Specifically, for n = 1 and λ = 1,

$$|Y_1^1| = B(\tfrac{1}{2}, \tfrac{1}{2}) = \textcolor{teal}{\pi}.$$

n-dimensional generalized Laguerre volume

$$\begin{align*} V_n &= \int_{\mathbb{R}^n} \left(\sqrt{x_1^2+\cdots+x_n^2}\right)^\alpha \exp\left(-\sqrt{x_1^2+\dots+x_n^2}\right)\\\ &= \begin{cases} 2\Gamma(1+\alpha)&\text{if $n=1$}\\\ \textcolor{orange}{\tau}\Gamma(2 + \alpha)&\text{if $n=2$}\\\ V_{n-2} \times \textcolor{orange}{\tau} (n+\alpha-1) (n+\alpha-2) / (n-2) & \text{otherwise} \end{cases} \end{align*}$$

The Gaussian integral

Compare

$$\int_{-\infty}^{\infty} \exp(-x^2)\,dx = \sqrt{\textcolor{teal}{\pi}},\qquad \int_{-\infty}^{\infty} \exp(-x^2 / 2)\,dx = \sqrt{\textcolor{orange}{\tau}}$$

One could argue that the latter is more "canonical" since it has variance and standard deviation of 1, not ½ and √½. Compare with the Normal distribution

$$\int_{-\infty}^{\infty} \frac{1}{\sigma \sqrt{\textcolor{orange}{\tau}}} \exp\left(-\frac{(x-\mu)^2}{2 \sigma^2}\right) \,dx= 1$$

Cauchy's integral formula

Let $U$ be an open subset of the complex plane $\mathbb{C}$, and suppose the closed disk $D$ defined as

$$D = \bigl\{z:|z-z_{0}|\leq r\bigr\}$$

is completely contained in $U$. Let $f:U\to\mathbb{C}$ be a holomorphic function, and let $\gamma$ be the circle, oriented counterclockwise, forming the boundary of $D$. Then for every $a$ in the interior of $D$,

$$f(a) = \frac{1}{\textcolor{orange}{\tau} i} \oint_{\gamma}\frac{f(z)}{z-a} dz.$$

Cauchy's residue theorem

Let $U$ be a simply connected open subset of the complex plane containing a finite list of points $a_1,\dots,a_n$, $U_0 = U \setminus \{a_1,\dots,a_n\}$, and a function f defined and holomorphic on $U_0$. Let γ be a closed rectifiable curve in $U_0$, and denote the winding number of $\gamma$ around $a_k$ by $I(\gamma, a_k)$. The line integral of $f$ around $\gamma$ is equal to $\textcolor{orange}{\tau} i$ times the sum of residues of $f$ at the points, each counted as many times as $\gamma$ winds around the point:

$$\oint_\gamma f(z)\,\mathrm{d}z = \textcolor{orange}{\tau} i \sum_{k=1}^n I(\gamma,a_k) \mathrm{Res}(f,a_k)$$

Fourier transform

$$\begin{align*} \hat{f}(\xi) &= \int_{-\infty}^{\infty} f(x) \exp(-i\textcolor{orange}{\tau} x\xi)\,dx,\\\ f(x) &= \int_{-\infty}^{\infty} \hat{f}(\xi) \exp(i\textcolor{orange}{\tau} x\xi)\,d\xi \end{align*}$$

nth roots of unity

$$z^n = 1 \quad\Rightarrow\quad z = \exp(i\textcolor{orange}{\tau} k / n) \quad\forall k=0,\dots,n-1$$

Euler's identity

$$\exp(i \textcolor{teal}{\pi}) + 1 = 0,\quad \exp(i \textcolor{orange}{\tau}) - 1 = 0$$

Stirling's approximation

$$n! \sim \sqrt{\textcolor{orange}{\tau} n} \left(\frac{n}{e}\right)^n$$

Particular values of the Gamma function

  • Positive half-integers:

    Γ(½) = √π, so all half-integer values of Γ contain that factor (recall Γ(z+1) = z Γ(z)):

    $$\Gamma(n + \tfrac{1}{2}) = \sqrt{\textcolor{teal}{\pi}} \prod_{i=0}^{n-1} \left(i + \frac{1}{2}\right)$$

    for n ∈ ℕ. Equivalently:

    $$\Gamma(n + \tfrac{1}{2}) = \sqrt{\textcolor{teal}{\pi}} \frac{(2n-1)!!}{2^n}$$

    or (for odd n > 0)

    $$\Gamma(\tfrac{n}{2}) = \sqrt{\textcolor{teal}{\pi}} \frac{(n-2)!!}{2^{(n-1)/2}}.$$

  • The multiplication theorem:

    $$\prod_{k=0}^{m-1}\Gamma\left(z+\frac{k}{m}\right) = \textcolor{orange}{\tau}^\frac{m-1}{2} m^{\frac{1}{2}-mz} \Gamma(mz),$$

    and its special case (m = 2), the Legendre duplication formula

    $$\Gamma(z)\Gamma(z+\tfrac{1}{2}) = 2^{1-2z}\sqrt{\textcolor{teal}{\pi}}\Gamma(2z).$$

  • Euler’s reflection formula:

    $$\Gamma(z)\Gamma(1-z) = \frac{\textcolor{teal}{\pi}}{\sin(\textcolor{teal}{\pi} z)}$$

Riemann zeta function

Values at even integers:

$$\zeta(2n) = \sum_{k=1}^{\infty} \frac{1}{k^{2n}} = (-1)^{n+1} \frac{\textcolor{orange}{\tau}^{2n} B_{2n}}{2 (2n)!},$$

for $n\in\mathbb{N}$, e.g.,

$$\zeta(2) = \sum_{k=1}^{\infty} \frac{1}{k^2} = \frac{\textcolor{teal}{\pi}^2}{6} = \frac{\textcolor{orange}{\tau}^2}{24}$$

A reflection formula:

$$\frac{\zeta(1-z)}{\zeta(z)} = 2 \frac{\Gamma(z)}{\textcolor{orange}{\tau}^z} \cos\left(\frac{\textcolor{orange}{\tau} z}{4}\right)$$

Weyl law

$$\lim_{\lambda\to\infty} \frac{N(\lambda)}{\lambda^{d/2}} = \textcolor{orange}{\tau}^{-d}\omega_d \mathrm{vol}(\Omega)$$

Error function

$$\mathrm{erf}(z) = \frac{2}{\sqrt{\textcolor{teal}{\pi}}} \int_0^z \exp(-t^2)\, dt.$$

In statistics, for non-negative values of x, the error function has the following interpretation: For a random variable Y that is normally distributed with mean 0 and standard deviation $1/\sqrt{2}$, erf(x) is the probability that Y falls in the range [−x, x]. The same property with standard deviation 1 is fulfilled by

$$\mathrm{erf}_1(z) = \mathrm{erf}(z / \sqrt{2}) = \frac{2}{\sqrt{\textcolor{orange}{\tau}}} \int_0^z \exp(-t^2 / 2)\, dt.$$

The sinc function and its power integrals

$$\int_{-\infty}^{\infty} \frac{\sin^n(x)}{x^n}\,dx = \frac{n \textcolor{orange}{\tau}}{2^n}\sum_{k=0}^{\lfloor n/2 \rfloor} \frac{(-1)^k (n-2k)^{n-1}}{k!(n-k)!}$$

for all $n\in\mathbb{N}$. Specifically,

$$\int_{-\infty}^{\infty} \frac{\sin(x)}{x}\,dx = \int_{-\infty}^{\infty} \frac{\sin^2(x)}{x^2}\,dx = \frac{\textcolor{orange}{\tau}}{2} = \textcolor{teal}{\pi}.$$

The Borwein integral

$$\int_{-\infty}^{\infty} \prod_{k=0}^n \frac{\sin(a_k x)}{a_k x}\,dx = \textcolor{teal}{\pi} C_n$$

with some rational $C_n$ (see here).


Physics

  • Cosmological constant:

    $$\Lambda = \frac{4\textcolor{orange}{\tau} G}{3c^2} \rho$$

  • Heisenberg's uncertainty principle:

    $$\Delta x \Delta p \ge \frac{h}{2 \textcolor{orange}{\tau}}$$

  • Einstein's field equation of general relativity:

    $$R_{\mu\nu} = \frac{4\textcolor{orange}{\tau} G}{c^4} T_{\mu\nu}$$

  • Coulomb's law for the electric force in vacuum:

    $$F = \frac{|q_1 q_2|}{2\textcolor{orange}{\tau} \varepsilon_0 r^2}$$

  • Magnetic permeability of free space:

    $$\mu_0 \approx 2\textcolor{orange}{\tau} \times 10^{-7} N/A^2$$

  • Approximate period of a simple pendulum with small amplitude:

    $$T \approx \textcolor{orange}{\tau} \sqrt{\frac{L}{g}}$$

  • Exact period of a simple pendulum with amplitude θ0:

    $$T = \frac{\textcolor{orange}{\tau}}{\mathrm{agm}(1, \cos(\theta_0/2))} \sqrt{\frac{L}{g}}$$

    (agm is the arithmetic-geometric mean.)

  • Kepler's third law of planetary motion:

    $$\frac{R^3}{T^2} = \frac{GM}{\textcolor{orange}{\tau}^2}$$

  • The buckling formula:

    $$F = \frac{\textcolor{orange}{\tau}^2 EI}{4L^2}$$

  • Reduced Planck constant:

    $$\hbar = \frac{h}{\textcolor{orange}{\tau}}$$

  • Reactance of an inductor:

    $$\textcolor{orange}{\tau} fL$$

  • Susceptance of a capacitor:

    $$\textcolor{orange}{\tau} fC$$

Quotes

  • Terence Tao (2007, here):

    It may be that 2πi is an even more fundamental constant than 2π or π. It is, after all, the generator of log(1). The fact that so many formulae involving πn depend on the parity of n is another clue in this regard.


  • John Conway (2008, from Constant Failure by Robert P Crease):

    [...] I posed this question to the Princeton University mathematician John Conway, one of the most creative mathematicians working today. Conway, it turned out, had strong feelings on the subject. “2π is obviously the correct constant!” he told me immediately — although he also told me of arguments, which he did not find persuasive, for a third option, π/2. [...]


  • Arthur Benjamin (2013, Twitter Q&A):

    I’m a big tau lover. I agree with the statement that if we could go back in time and change the factor to tau we would have simplified our theorems and formulas. Obviously, it will be very hard to change people’s perceptions in order to use tau, but maybe in mathematics there is enough of a will to do such a thing. I’ve seen books now that proudly claim “tau certified”.


  • John Baez (2022, from 12 numbers that are cooler than pi):

    Using τ makes every formula clearer and more logical than using π. Our focus on π rather than 2π is a historical accident.


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The better pi

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